1,071 research outputs found
Spectral gaps for periodic Schr\"odinger operators with strong magnetic fields
We consider Schr\"odinger operators
with the periodic magnetic field on covering spaces of
compact manifolds. Under some assumptions on , we prove that there are
arbitrarily large number of gaps in the spectrum of these operators in the
semiclassical limit of strong magnetic field .Comment: 14 pages, LaTeX2e, xypic, no figure
Magnetic calculus and semiclassical trace formulas
The aim of these notes is to show how the magnetic calculus developed in
\cite{MP, IMP1, IMP2, MPR, LMR} permits to give a new information on the nature
of the coefficients of the expansion of the trace of a function of the magnetic
Schr\"odinger operator whose existence was established in \cite{HR2}
Agmon-type estimates for a class of jump processes
In the limit epsilon to 0 we analyze the generators H_epsilon of families of
reversible jump processes in R^d associated with a class of symmetric non-local
Dirichlet-forms and show exponential decay of the eigenfunctions. The
exponential rate function is a Finsler distance, given as solution of a certain
eikonal equation. Fine results are sensitive to the rate function being C^2 or
just Lipschitz. Our estimates are analog to the semi-classical Agmon estimates
for differential operators of second order. They generalize and strengthen
previous results on the lattice epsilon Z^d. Although our final interest is in
the (sub)stochastic jump process, technically this is a pure analysis paper,
inspired by PDE techniques
Applications of Magnetic PsiDO Techniques to Space-adiabatic Perturbation Theory
In this review, we show how advances in the theory of magnetic
pseudodifferential operators (magnetic DO) can be put to good use in
space-adiabatic perturbation theory (SAPT). As a particular example, we extend
results of [PST03] to a more general class of magnetic fields: we consider a
single particle moving in a periodic potential which is subjectd to a weak and
slowly-varying electromagnetic field. In addition to the semiclassical
parameter \eps \ll 1 which quantifies the separation of spatial scales, we
explore the influence of additional parameters that allow us to selectively
switch off the magnetic field.
We find that even in the case of magnetic fields with components in
, e. g. for constant magnetic fields, the results of
Panati, Spohn and Teufel hold, i.e. to each isolated family of Bloch bands,
there exists an associated almost invariant subspace of and an
effective hamiltonian which generates the dynamics within this almost invariant
subspace. In case of an isolated non-degenerate Bloch band, the full quantum
dynamics can be approximated by the hamiltonian flow associated to the
semiclassical equations of motion found in [PST03].Comment: 32 page
- …